Bayesian Estimators Using Record Statistics of ... - CiteSeerX

DOI: http://dx.doi.org/10.5351/CKSS.2012.19.3.479

Communications of the Korean Statistical Society 2012, Vol. 19, No. 3, 479–493

Bayesian Estimators Using Record Statistics of Exponentiated Inverse Weibull Distribution Yongku Kima , Jung In Seo a , Suk-Bok Kang 1, a a

Department of Statistics, Yeungnam University

Abstract The inverse Weibull distribution(IWD) is a complementary Weibull distribution and plays an important role in many application areas. In this paper, we develop a Bayesian estimator in the context of record statistics values from the exponentiated inverse Weibull distribution(EIWD). We obtained Bayesian estimators through the squared error loss function (quadratic loss) and LINEX loss function. This is done with respect to the conjugate priors for shape and scale parameters. The results may be of interest especially when only record values are stored.

Keywords: Bayesian estimation, exponentiated inverse Weibull distribution, record statistics.

1. Introduction The probability density function(pdf) and cumulative distribution function(cdf) of the random variable X having the exponentiated inverse Weibull distribution are given by f (x; α, β, γ) =

) ( αγ exp −α(βx)−γ x−γ−1 βγ

(1.1)

and ) ( F(x; α, β, γ) = exp −α(βx)−γ ,

x > 0, α, β, γ > 0.

The kth moment of this distribution that was introduced by Ali et al. (2007) is ) ( ) α γk ( k E Xk = k Γ 1 − , γ > k. γ β

(1.2)

(1.3)

Therefore, the mean and the variance of the exponentiated inverse Weibull distribution can be written as follows. ( ) 1 αγ 1 E (X) = Γ 1− (1.4) β γ and 2

αγ Var (X) = 2 β 1

 ( ) { ( )}2    2 1 Γ 1 −  , − Γ 1− γ γ 

for γ > 2.

Corresponding author: Professor, Department of Statistics, Yeungnam University, Gyeongsan, 712-749, Korea. E-mail: [email protected]

(1.5)

480

Yongku Kim, Jung In Seo, Suk-Bok Kang

Both the mean (1.4) and the variance (1.5) increase as α increases, when γ > 2. From (1.2), the reliability function of the exponentiated inverse Weibull distribution is given by ( ) R(t) = 1 − F(t) = 1 − exp −α(βt)−γ , t > 0. (1.6) Note that the inverse Weibull distribution is a special case of (1.1) when α = 1. The inverse Weibull distribution is the complementary Weibull distribution and plays an important role in many applications including the dynamic components of diesel engines, the times to breakdown of an insulating fluid subject to the action of constant tension and flood data (see Nelson, 1982; Maswadah, 2003). In addition, it has been used quite extensively when the data indicate a monotone hazard function because of the flexibility of the pdf and its corresponding hazard function. Studies for the inverse Weibull distribution have been conducted by many authors. Calabria and Pulcini (1994) studied Bayes 2-sample prediction for the inverse Weibull distribution. Mahmoud et al. (2003) considered the order statistics arising from the inverse Weibull distribution and derived the exact expression for the single moments of order statistics. They also obtained variances and covariances based on the moments of order statistics. Chandler (1952) introduced the study of record values and documented many of the basic properties of records. Record values arise in many real-life situations that involve the weather, sports, economics and life tests. Record model is related to the order statistics model, both of which appear in many statistical applications and are widely used in statistical modeling and inference because it can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. In particular, Balakrishnan et al. (1992) established some recurrence relations for the single and double moments of lower record values from Gumble distribution. Soliman et al. (2006) obtained Bayes estimators based on record statistics for two unknown parameters of the Weibull distribution. Recently, Sultan (2008) derived the Bayes estimators and obtained the estimators of the reliability and hazard functions for the unkonwn parameters of the inverse Weibull distribution based on lower record values. The squared error loss function(SELF) is a symmetric loss function assigning equal losses to overestimation and underestimation. Therefore, under the SELF, Bayes estimator is defined by the posterior expectation. However, such a restriction may be impractical because an overestimate is usually more serious than an underestimate in the estimation of reliability and failure rate functions. In this case the use of a symmetrical loss function might be inappropriate. To cover this drawback, we consider two types asymmetric loss functions known as the LINEX loss function(LLF) and the SQUAREX loss function(SLF). The LLF was introduced by Varian (1975) and received significant popularity due to Zellner (1986). It may be expressed as L(△) ∝ exp(c△) − c△ − 1, c , 0, where △ = θˆ − θ and θˆ is an estimator of θ. The sign and magnitude of the shape parameter c represents the direction and degree of symmetry, respectively. When c is positive, the overestimation is more serious than underestimation and the situation is reverse when c is negative. For c = 1, the LLF is quite asymmetric (about zero) with overestimation being more costly than underestimation. If c is close to zero, the LINEX loss is approximately the squared error loss and therefore almost symmetric. By Zellner (1986), the Bayes estimator of θ under the LLF was given by θˆ L = −(1/c) log[Eπ (e−cθ )], provided that the expectation exists and is finite. Secondly, the SLF proposed by Thompson and Basu (1996), is a generalization of LLF. It has the following form. L(△) ∝ exp(c△) + d△2 − c△ − 1, d > 0, c and △ are as before. Hence, if d = 0, the SLF is identical with the LLF; if c = 0, it reduces to the SELF. Under the SLF, the Bayes estimator of θ is θˆS L = θˆ L + (1/c) log[1 + (2d/c)(θˆ s − θˆS L )]. The exponentiated inverse Weibull distribution is the most attractive generalization of inverse

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Weibull distribution and provides a better fit for real life data compared to an inverse Weibull distribution. In this paper, we develop a Bayesian estimator in the context of record statistics values from the EIWD. We obtained Bayesian estimators using the squared error loss function (quadratic loss) and LINEX loss function. This is done with respect to the conjugate priors for the shape and scale parameters. The outline of the remaining sections is as follows. In Section 2, we develop the exact form of the single moment and the maximum likelihood estimators(MLEs) of lower record values from the EIWD. Section 3 details Bayesian estimation in the context of record statistics values from the EIWD under three types loss functions. In Section 4, we also analyze application examples to illustrate the application of different derived estimators. Finally, in the estimated risks, the Bayes estimators are compared with MLEs through Monte Carlo simulations.

2. Maximum Likelihood Estimation In this section, we consider the MLEs of the unknown parameters and reliability function R(t) in an exponentiated inverse Weibull distribution based on lower record values. Let X1 , X2 , X3 , . . . be a sequence of independent and identically distributed(iid) random variables with cdf F(x) and pdf f (x). Setting Yn = min(X1 , X2 , . . . , Xn ), n ≥ 1, we say that X j is a lower record and denoted by XL( j) if Y j < Y j−1 , j > 1. The indices at which the lower record values occur are given by the record times {L(n), n ≥ 1}, where L(n) = min{ j| j > L(n − 1), X j < XL(n−1) }, n > 1, with L(1) = 1. The corresponding likelihood function of the first n lower record values, xL(1) , . . . , xL(n) is n−1 ∏ f (xL(i) ) L = f (xL(n) ) . F(x L(i) ) i=1

(2.1)

Suppose we observe n lower record values xL(1) , . . . , xL(n) from the exponentiated inverse Weibull distribution with pdf (1.1). It follows, from (1.1), (1.2), and (2.1), that ( )∏ ( )n n α αγ L(α, β, γ) = γ exp − x−γ−1 . (2.2) β (βxL(n) )γ i=1 L(i) As a property of lower record values, its kth moment can be obtained by ( ) α γk Γ(n − k/γ) k E XL(n) = k , Γ(n) β

γ > k.

(2.3)

Now, we derive the MLEs of the parameters of the exponentiated inverse Weibull distribution when record values are given as data. From (2.2), the natural logarithm of the likelihood function is given by log L(α, β, γ) = n log α − nγ log β + n log γ −

α (βxL(n) )γ

− (γ + 1)

n ∑

log xL(i) .

(2.4)

i=1

From the log-likelihood function (2.4), we obtain the likelihood equations for α, β, and γ as ( )γ ∂ log L n 1 = − = 0, ∂α α βxL(n) ( )γ γn αγ 1 ∂ log L =− − = 0, ∂β β β βxL(n)

(2.5) (2.6)

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and ( ) ∑ n α log βxL(n) ∂ log L n = − n log β + ( log xL(i) = 0. )γ − ∂γ γ βxL(n) i=1

(2.7)

By solving the above equations, we can find the following MLEs of the unknown parameters α, β, and γ. ( )γˆ ˆ L(n) , (2.8) αˆ = n βx 1 ( ) γˆ αˆ βˆ = x−1 (2.9) L(n) , n and γˆ =

n log βˆ +

n

∑n i=1

( )−ˆγ ( ). ˆ L(n) log βx ˆ L(n) log xL(i) − αˆ βx

(2.10)

The MLE γˆ in (2.10), in conjunction with the MLE βˆ in (2.9), reduces to n . log x L(i) − n log xL(n) i=1

γˆ = ∑n

(2.11)

By the invariance property of the MLE, we can obtain the MLE of reliability function R(t) to be     α ˆ  ˆ = 1 − exp − ( )  . (2.12) R(t)  γˆ  ˆ  βt

3. Bayesian Estimation In this section, we estimate α, β, γ, and R(t), through consideration of symmetric loss function and two types of asymmetric loss functions and discuss method to obtain hyperparameters.

3.1. Unknown parameter α Under the assumption that parameters β and γ are known, a natural conjugate prior for the parameter α is a gamma prior as follows. π(α) =

ba a−1 −αb α e , Γ(a)

α > 0, a, b > 0.

(3.1)

It follows, from (3.1), that the posterior distribution of α is given by ( ) b + (βxL(n) )−γ n+a n+a−1 −α(b+(βxL(n) )−γ ) π(α|x) = α e , (3.2) Γ(n + a) ( ) which is a Gamma n + a, b + (βxL(n) )−γ . From (3.2), the Bayes estimators of α and R(t) based on the SELF can be derived, respectively, as ∫ ∞ n+a (3.3) αˆ s = απ(α|x)dα = b + (βxL(n) )−γ 0

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and ∫ Rˆ s (t) =

∞

( R(t)π(α|x)dα = 1 − 1 +

0

(βt)−γ b + (βxL(n) )−γ

)−(n+a) .

(3.4)

Likewise, the Bayes estimators of α and R(t) based on the LLF can be derived, respectively, as ( ) ∫ ∞ 1 c n+a αˆ L = − log e−cα π(α|x)dα = log 1 + (3.5) c c b + (βxL(n) )−γ 0 and 1 Rˆ L (t) = − log c

∫

∞

e−cR(t) π(α|x)dα

0

 ( )−(n+a)  ∞ m −γ  −c ∑  1 c m(βt)  = − log e 1+ −γ c m! b + (βxL(n) ) m=0 ∞ ( )−(n+a)  ∑ cm  1 m(βt)−γ  . = 1 − log  1+ −γ c m! b + (βxL(n) ) m=0

Also, the Bayes estimators of α and R(t) based on the SLF are [ ] 1 2d (αˆ s − αˆ S L ) αˆ S L = αˆ L + log 1 + c c

(3.6)

(3.7)

and [ )] 1 2d ( ˆ ˆ ˆ ˆ RS L (t) = RL (t) + log 1 + R s (t) − RS L (t) . c c

(3.8)

Since the Bayes estimators (3.7) and (3.8) are not explicit form, we can solve through the use of a numerical method such as Newton-Raphson. To assess the performance of the MLEs and the Bayes estimators, we simulate the estimated risks of all derived estimators through a Monte Carlo simulation method when the parameters β and γ are known. The following procedure is required to obtain estimated risks. After setting E(α) and Var(α) from the prior density (3.1), we obtain the hyperparameters a and b of the gamma prior (3.1) by solving them. Note that E(α) is the actual value for α. We generate the lower record values from the exponentiated inverse Weibull distribution with α = E(α). By using these values, we can finally obtain the Bayes estimators. The estimated risks for each estimator are calculated as the average of their squared deviations for 10,000 repetitions. It is expressed as )2 1 ∑( θt − θˆ . n i=1 n

Here θt and θˆ is the actual value and the estimate of θ, respectively. The estimated risks of α and R(t) are given in Table 1. The Bayes estimators based on the SELF, the LLF, and the SLF are denoted by BS, BL, and BSL, respectively.

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3.2. Unknown parameters (α, β) In the case of the two parameters problem, we need to specify a general joint prior for α and β that may lead to computational complexities. To avoid this problem, we consider Soland’s method. Soland (1969) considered a family of joint prior distribution that places continuous distribution on the scale parameter and discrete distributions on the shape parameter. Suppose that β is restricted to the values β1 , β2 , . . . , β J with prior probabilities η1 , η2 , . . . , η J , that is, π(β j ) = P[β = β j ] = η j ,

j = 1, 2, . . . , J.

(3.9)

Further, suppose that the conditional α upon β = β j has a natural conjugate prior as gamma(a j , b j ) with pdf a

π(α|β = β j ) =

bjj Γ(a j )

αa j −1 e−αb j ,

α > 0, a j , b j > 0.

(3.10)

Combining (2.2) and (3.10), we get the conditional posterior of α|β = β j as ( π(α|β = β j , x) =

b j + (β j xL(n) )−γ Γ(n + a j )

)n+a j −γ αn+a j −1 e−α(b j +(β j xL(n) ) )

(3.11)

which is a Gamma(n + a j , b j + (β j xL(n) )−γ ). In view of the discrete version of Bayes theorem, we obtain the marginal posterior of β as ∫ ∞ π M (β j |x) ∝ L(α, β)π(α|β = β j )π(β j )dα 0

∏ a η j b j j Γ(n + a j )γn ni=1 x−γ−1 L(i) = ( )n+a j . γn Γ(a j )β j b j + (β j xL(n) )−γ

(3.12)

Hence, we get ∏ a η j b j j Γ(n + a j )γn ni=1 x−γ−1 L(i) π M (β j |x) = G(β) ( )n+a j , γn −γ Γ(a j )β j b j + (β j xL(n) )

(3.13)

where G(β) is the normalizing constant given by ∏ a J ∑ η j b j j Γ(n + a j )γn ni=1 x−γ−1 L(i) G (β) = ( )n+a j . γn −γ j=1 Γ(a j )β j b j + (β j xL(n) ) −1

(3.14)

Multiplying (3.11) by (3.13), we also obtain the marginal posterior of α as π M1 (α|x) =

J ∑ j=1

π(α|β = β j , x)π M (β j |x).

(3.15)

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From (3.13) and (3.15), the Bayes estimators of α, β, and R(t) based on the SELF are derived, respectively, as ∫ ∞ ∫ ∞ J ∑ αˆ s = απ M1 (α|x)dα = π M (β j |x) απ(α|β = β j , x)dα 0

=

J ∑

π M (β j |x)

Γ(n + a j )

j=1

=

J ∑

π M (β j |x)

j=1

βˆ s =

0

j=1

( )n+a j ∫ b j + (β j xL(n) )−γ

J ∑

∞

−γ

e−α(b j +(β j xL(n) ) ) αn+a j dα

0

n + aj , b j + (β j xL(n) )−γ

β j π M (β j |x),

(3.16)

(3.17)

j=1

and ∫

∫

( ) −γ π M1 (α|x) 1 − e−α(β j t) dα 0 0 ( )n+a j ∫ ∞ J ∑ ( ) b j + (β j xL(n) )−γ −γ −γ e−α(b j +(β j xL(n) ) ) αn+a j −1 1 − e−α(β j t) dα × = Γ(n + a j ) 0 j=1 ( )−(n+a j ) J ∑ (β j t)−γ =1− π M (β j |x) 1 + . b j + (β j xL(n) )−γ j=1

Rˆ s (t) =

∞

π M1 (α|x)R(t)dα =

∞

The Bayes estimator of a function g(α, β) based on the LLF is given by [ ( )] 1 gˆ (α, β) = − log E e−cg(α, β) c

(3.18)

(3.19)

which can be written as

  J ∫  ∑ 1 −cg(α, β) π(α|β = β j , x)dα . gˆ (α, β) = − log  π M (β j |x) e c j=1

(3.20)

By using (3.20), the Bayes estimators of α, β, and R(t) based on the LLF are derived, respectively, as ∫ ∞ 1 αˆ L = − log π M1 (α|x)e−cα dα c 0  J  ( −γ )n+a j  ∑  b + (β x ) 1 j j L(n)  = − log  π M (β j |x) −γ c b j + (β j xL(n) ) + c j=1  J ( )−(n+a j )  ∑  1 c  , (3.21) = − log  π M (β j |x) 1 + −γ c b j + (β j xL(n) ) j=1   J  ∑ 1 −cβ  j (3.22) βˆ L = − log  π M (β j |x)e  , c j=1

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and 1 Rˆ L (t) = − log c

∫

∞

π M1 (α|x)e−cR(t) dα

0

 ( )−(n+a j )  J ∑ ∞ −γ m  ∑  m(β t) 1 c j  = − log e−c π M (β j |x) 1+ −γ c m! b + (β x ) j j L(n) j=1 m=0  J ∞ ( )−(n+a j )  −γ m ∑ ∑  m(β t) 1 c j  = 1 − log  π M (β j |x) 1+ −γ c m! b + (β x ) j j L(n) j=1 m=0

(3.23)

because of ( ) ∞ ( ) ∑ cm αm −γ exp ce−α(βt) = exp − . m! (βt)γ m=0

(3.24)

On the basis of the Bayes estimators based on the SELF and the LLF, we can find the following Bayes estimators based on the SLF. [ ] 1 2d (αˆ s − αˆ S L ) , αˆ S L = αˆ L + log 1 + (3.25) c c [ ] ) 1 2d ( ˆ βˆ S L = βˆ L + log 1 + β s − βˆ S L , (3.26) c c and [ )] 1 2d ( ˆ Rˆ S L (t) = Rˆ L (t) + log 1 + R s (t) − Rˆ S L (t) . c c

(3.27)

As mentioned in the Section 3.1, αˆ S L , βˆ S L , and Rˆ S L (t) are obtained by applying the numerical method.

3.3. Unknown parameters (α, β, γ) For the same reason, we expand the method employed by Soland (1969). Suppose that β and γ are restricted to a finite number of values β1 , β2 , . . . , β J and γ1 , γ2 , . . . , γK with prior probabilities η1 , η2 , . . . , η J and ζ1 , ζ2 , . . . , ζK , respectively. That is, π(β j ) = P[β = β j ] = η j ,

j = 1, 2, . . . , J

(3.28)

k = 1, 2, . . . , K.

(3.29)

and π(γk ) = P[γ = γk ] = ζk ,

Now, assume that the conditional α upon β = β j and γ = γk , j = 1, 2, . . . , J and k = 1, 2, . . . , K has a gamma (a jk , b jk ) prior with pdf a

π(α|β = β j , γ = γk ) =

b jkjk Γ(a jk )

αa jk −1 e−αb jk ,

α > 0, a jk , b jk > 0.

(3.30)

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Then, the conditional posterior of α|β = β j , γ = γk and the marginal joint posterior of (β, γ) can be obtained by ( )n+a jk ( ) b jk + (β j xL(n) )−γk −γk αn+a jk −1 e−α(b jk +(β j xL(n) ) ) (3.31) π α|β = β j , γ = γk , x = Γ(n + a jk ) and ∏ a k −1 η j ζk b jkjk Γ(n + a jk )γkn ni=1 x−γ L(i) π M (β j , γk |x) = G(β, γ) ( )n+a jk , Γ(a jk )βγj k n b jk + (β j xL(n) )−γk

(3.32)

where G(β, γ) is the normalizing constant given by ∏n −γk −1 J ∑ K η ζ ba jk Γ(n + a )γn ∑ jk k j k jk i=1 xL(i) G (β, γ) = ( )n+a jk . γk n b jk + (β j xL(n) )−γk j=1 k=1 Γ(a jk )β j −1

(3.33)

Note that α|β = β j , γ = γk has a Gamma(n + a jk , b jk + (β j xL(n) )−γk ). Using (3.31) and (3.32), we can obtain the marginal posterior of α as π M2 (α|x) =

J ∑ K ∑ ( ) ( ) π α|β = β j , γk , x π M β j , γk |x .

(3.34)

j=1 k=1

From (3.32) and (3.34), the Bayes estimators of α, β, γ, and R(t) based on the SELF are derived, respectively, as ∫

∞

αˆ s =

απ M2 (α|x)dα =

0

=

J ∑ K ∑

π M (β j , γk |x)

J ∑ K ∑

∞

π M (β j , γk |x)

απ(α|β = β j , γ = γk , x)dα

0

j=1 k=1

j=1 k=1

βˆ s =

∫

J ∑ K ∑

n + a jk , b jk + (β j xL(n) )−γk

(3.35)

β j π M (β j , γk |x),

(3.36)

γk π M (β j , γk |x),

(3.37)

j=1 k=1

γˆ s =

J ∑ K ∑ j=1 k=1

and ∫

∞

Rˆ s (t) =

∫

0

=

J ∑ K ∑

∫ π M (β j , γk |x)

j=1 k=1

=1−

∞

π M2 (α|x)R(t)dα =

J ∑ K ∑ j=1 k=1

( ) −γk π M2 (α|x) 1 − e−α(β j t) dα

0 ∞

( ) −γk π(α|β = β j , γ = γk , x) 1 − e−α(β j t) dα

0

( π M (β j , γk |x) 1 +

(β j t)−γk b jk + (β j xL(n) )−γk

)−(n+a jk ) .

(3.38)

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Similarly, the Bayes estimators of α, β, γ, and R(t) based on the LLF are obtained by ∫ ∞ 1 αˆ L = − log π M2 (α|x)e−cα dα c 0  J K ( )−(n+a jk )  ∑ ∑  1 c  , = − log  π M (β j , γk |x) 1 + −γ c b jk + (β j xL(n) ) k j=1 k=1  J K  ∑ ∑  1 −cβ  βˆ L = − log  π M (β j , γk |x)e j  , c j=1 k=1  J K  ∑ ∑  1 −cγ  k γˆ L = − log  π M (β j , γk |x)e  , c j=1 k=1 and 1 Rˆ L (t) = − log c

∫

∞

(3.40)

(3.41)

π M2 (α|x)e−cR(t) dα

0

 ( )−(n+a jk )  J ∑ K ∑ ∞ −γk m  ∑  m(β t) c 1 j −c  π M (β j , γk |x) = − log e 1+ −γk c m! b + (β x ) jk j L(n) j=1 k=1 m=0  J K ∞ ( )−(n+a jk )  −γk m ∑ ∑ ∑  m(β t) c 1 j  . π M (β j , γk |x) = 1 − log  1+ −γ c m! b jk + (β j xL(n) ) k j=1 k=1 m=0

Finally, the Bayes estimators of α, β, γ, and R(t) based on the SLF are given by [ ] 1 2d (αˆ s − αˆ S L ) , αˆ S L = αˆ L + log 1 + c c [ )] 1 2d ( ˆ βˆ S L = βˆ L + log 1 + β s − βˆ S L , c c [ ] 1 2d (ˆγ s − γˆ S L ) , γˆ S L = γˆ L + log 1 + c c and

(3.39)

[ )] 1 2d ( ˆ Rˆ S L (t) = Rˆ L (t) + log 1 + R s (t) − Rˆ S L (t) . c c

(3.42)

(3.43) (3.44) (3.45)

(3.46)

In order to apply the methods discussed in this section, we should first extract the values of (β j , η j ), (γk , ζk ) and the hyperparameters (a jk , b jk ) in the conjugate prior (3.30). For each choice of (a jk , b jk ), it is difficult to find the prior of α conditioned on each value of β j and γk . An alternative method to obtain the values (a jk , b jk ) can be based on the expected value of the reliability function R(t) conditional on β = β j and γ = γk , which is given using (3.30) by a b jkjk ∫ ∞ ( ( )) [ ] 1 − exp −α(β j t)−γk αa jk−1 e−αb jk dα E R(t)|β = β j , γ = γk = Γ(a jk ) 0 ( ) (β j t)−γk −a jk =1− 1+ , t > 0. (3.47) b jk

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Table 1: The maximum flood level over a 20 four–year period (1890–1969) 0.654 0.269

0.613 0.740

0.315 0.418

0.449 0.412

0.297 0.494

0.402 0.416

0.379 0.338

0.423 0.392

0.379 0.484

0.324 0.265

If we are able to specify two values (t1 , R(t1 )) and (t2 , R(t2 )) from prior beliefs about the distribution, the values of a jk and b jk can be obtained numerically from (3.47). Otherwise, a nonparametric procedure can be used to estimate the corresponding two different values of R(t). We use mid-point estimator for R(t) as a nonparametric method.

4. Application We present two examples to illustrate the methods of inference discussed in the previous sections.

4.1. Real data Consider the real data given by Dumonceaux and Antle (1973) which represent the maximum flood level (in millions of cubic feet per second) of the Susquehenna River at Harrisburg, Pennsylvania over a 20 four–year period (1890–1969). This data given in Table 1 has been utilized by some authors such as Maswadah (2003) and Sultan (2008). Maswadah (2003) showed that this real data follow an inverse Weibull distribution giving a rough indication of the goodness of fit for the model. During this period, 6 lower records of the maximum flood level are observed, they are 0.654,

0.613,

0.315,

0.297,

0.269,

0.265.

In this example, we use gamma prior for the parameter α and discrete priors for the parameters β and γ. The values of β j , γk and the hyperparameters of the gamma prior (3.30) are derived by the following steps. First, we estimate two values of the reliability function using the mid-point estimator for R(ti = xL(i) ) = (n−i+0.5)/n, i = 1, 2, . . . , n. Here, we assume that the reliability for t1 = 0.613 and t2 = 0.269 are, respectively, R(t1 ) = 0.25, and R(t2 ) = 0.75. Next, we obtain the MLE γˆ = 2.93565 from (2.11) based on the above 6 lower record values when β = 1. Finally, we assume that γk = 2.6(0.1)3.2 and β j = 0.8(0.1)1.2. Therefore, the values of the hyperparameters a jk and b jk at each value of β j and γk are obtained by solving the following equations using Newton-Raphson method. ( ) (β j 0.613)−γk −a jk 1− 1+ = 0.25 b jk

(4.1)

and (

(β j 0.269)−γk 1− 1+ b jk

)−a jk = 0.75.

(4.2)

Table 2 shows the values of the hyperparameters and posterior probabilities obtained for each β j and γk . By using entries of Table 2, the ML estimates, and the Bayes estimates of α, β, γ, and R(t) are calculated. The results are given in Table 3. Note that the positive value of c is considered here because overestimation is more serious than underestimation in this example. We see that the MLEs are nearly equal to the Bayes estimates. To check the goodness of fit for the exponentiated inverse ˆ and γˆ , we conduct a simple test. A simple plot of 6 lower records of Weibull distribution with α, ˆ β, the maximum flood level against the expected values of the first exponentiated inverse Weibull lower record values indicate a very strong correlation (0.895). Besides, we have nearly the same results for

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Table 2: Prior information, hyperparameters and the posterior probabilities j (η j = 1/5)

βj

1

0.8

2

0.9

3

1.0

4

1.1

5

1.2

k (ξk = 1/7) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7

γk 2.6 2.7 2.8 2.9 3.0 3.1 3.2 2.6 2.7 2.8 2.9 3.0 3.1 3.2 2.6 2.7 2.8 2.9 3.0 3.1 3.2 2.6 2.7 2.8 2.9 3.0 3.1 3.2 2.6 2.7 2.8 2.9 3.0 3.1 3.2

a jk 1.087650 0.963320 0.866560 0.789100 0.725400 0.672210 0.626990 1.087670 0.963390 0.866620 0.789070 0.725440 0.672230 0.627070 1.087550 0.963320 0.866610 0.789040 0.725470 0.672240 0.627030 1.087570 0.963320 0.866710 0.789140 0.725490 0.672170 0.627140 1.087660 0.963380 0.866710 0.789080 0.725530 0.672220 0.627080

b jk 21.060400 19.674200 18.673900 17.950100 17.419000 17.045500 16.791200 15.505300 14.316900 13.429800 12.755200 12.235600 11.832200 11.521700 11.787600 10.770700 9.998600 9.396400 8.920400 8.535700 8.223100 9.200600 8.327000 7.658500 7.129100 6.702600 6.350600 6.064100 7.339000 6.584200 6.002500 5.538300 5.163300 4.850000 4.589200

π jk 0.027100 0.028722 0.029543 0.029683 0.029269 0.028424 0.027258 0.027100 0.028721 0.029543 0.029683 0.029269 0.028424 0.027259 0.027101 0.028722 0.029543 0.029683 0.029269 0.028424 0.027258 0.027101 0.028722 0.029543 0.029683 0.029270 0.028423 0.027260 0.027100 0.028722 0.029543 0.029683 0.029270 0.028424 0.027259

the Bayes estimates of α, β, and γ. Therefore, the assumption that these record values are from the exponentiated inverse Weibull distribution seems quite reasonable. The data given in Table 4 are the ˆ expected values of the first exponentiated inverse Weibull lower record values with the MLEs α, ˆ β, and γˆ .

4.2. Simulation study Similarly, we also consider simulated data consisting of 6 record values from an exponentiated inverse Weibull distribution. A set of lower record values is generated from the standard exponentiated inverse Weibull distribution with α = 0.5 and γ = 2.5. The actual generated population values of R(t = 0.5) is 0.94089. The following are the simulated 6 lower record values. 1.20300,

1.06597,

0.47784,

0.40798,

0.40367,

0.38334.

Using the formulae presented in Section 2 and Section 3, we obtain the Bayes and ML estimates of α, β, γ, and R(t). For α and γ, the Bayes estimates are closer to the actual values. Specially, the closest estimate to the actual value is asymmetric Bayes estimate under SLF with c = 1.5 and d = 3.0. We

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491

Table 3: Estimates of α, β, γ, and R(t = 0.5) for real data MLE

BS

α

0.12162

0.13884

β

1.00000

1.00000

γ

2.93565

2.90122

R(t)

0.60565

0.58662

BL (BSL) c = 1.5 (d = 0.5) 0.13310 (0.13486) 0.98507 (0.98966) 2.87179 (2.88081) 0.56982 (0.57498)

c = 0.5 (d = 0.5) 0.13684 (0.13844) 0.99500 (0.99900) 2.89134 (2.89925) 0.58106 (0.58551)

c = 2.5 (d = 0.5) 0.12965 (0.13092) 0.97533 (0.97872) 2.85291 (2.85953) 0.55847 (0.56234)

Table 4: Expected values and real data for the simple plot (i ) E XL(i) Real Data

1 0.66710 0.654

2 0.43986 0.613

3 0.36494 0.315

4 0.32350 0.297

5 0.29595 0.269

6 0.27579 0.265

Table 5: Estimates of α, β, γ, and R(t = 0.5) for simulated data MLE

BS

α

0.60122

0.51592

β

1.00000

1.00000

γ

2.39933

2.58071

R(t)

0.95807

0.91231

c = 0.5 (d = 3.0) 0.49625 (0.51558) 0.99500 (0.99983) 2.57109 (2.58043) 0.91050 (0.91224)

BL (BSL) c = 1.0 (d = 3.0) 0.47863 (0.51094) 0.99002 (0.99860) 2.56156 (2.57808) 0.90858 (0.91778)

c = 1.5 (d = 3.0) 0.46277 (0.50130) 0.98507 (0.99592) 2.55229 (2.57293) 0.90657 (0.91074)

also see that the asymmetric Bayes estimates under SLF draw closer to the actual values as c increases; however, the MLEs are the closest to the actual values for β and R(t). These values are given in Table 5. In general, it is difficult to judge which one is better estimator through a set of sample. A simulation study is conducted to see the efficiency of the Bayes and ML estimation methods in terms of estimated risks. The estimated risks for each estimator are calculated as the average of their squared deviations for 10,000 repetitions according to method discussed in Chapter 4. Samples of lower record values with size n = 10, are generated from the exponentiated inverse Weibull distribution with α = 0.05, β = 0.6, and γ = 1.2. For β = 0.6 and γ = 1.2, we consider the prior over the interval (0.1, 1.0) and (0.7, 1.6) by the discrete priors with β and γ taking the 10 values, each with probability 0.1. To obtain the Bayes estimates, we first calculate two values of the reliability function R(t2 = xL(2) ) and R(t9 = xL(9) ) and then can obtain the hyperparameters a jk and b jk using the expected value of the R(t) in (3.47). The posterior probabilities are easily calculated from a jk and b jk at each value of β j and γk . Through these steps, we obtain the Bayes estimates. By 10,000 repeating this procedure, the estimated risks for α, β, γ, and R(t) are obtained. For α = 0.05, β = 2, and γ = 1.5, the same simulation method is carry out. The results are presented in Table 6. From the table, we can see that the Bayes estimators are generally better than their corresponding MLEs. For α and R(t), the Bayes estimators relative to asymmetric loss function are more efficient than the Bayes estimators under symmetric loss function such as SELF. In addition, the estimated risks of them decrease as c increases for fixed d.

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Table 6: The estimated risks of α, β, γ, and R(t) when record values of size is 10 Actual values: (α, β, γ, R(t = 0.5)) = (0.05, 0.6, 1.2, 0.19107) MLE

BS

α

0.51779

0.21478

β

0.02719

0.00250

γ

0.20427

0.07541

R(t)

0.41108

0.27623

c = 1.0 (d = 0.05) 0.15282 (0.15798) 0.00826 (0.00760) 0.08267 (0.08197) 0.26283 (0.26403)

BL (BSL) c = 3.0 (d = 0.05) 0.09004 (0.09111) 0.027400 (0.02698) 0.09711 (0.09684) 0.23540 (0.23583)

c = 5.0 (d = 0.05) 0.05991 (0.06033) 0.05048 (0.05017) 0.11086 (0.11069) 0.20828 (0.20852)

c = 1.0 (d = 0.05) 0.12347 (0.12553) 0.01226 (0.01170) 0.07915 (0.07832) 0.07190 (0.07226)

BL (BSL) c = 3.0 (d = 0.05) 0.09815 (0.09884) 0.02196 (0.02150) 0.09196 (0.09157) 0.06673 (0.06694)

c = 5.0 (d = 0.05) 0.08557 (0.08609) 0.03307 (0.03268) 0.11086 (0.09993) 0.06204 (0.062176)

Actual values: (α, β, γ, R(t = 0.5)) = (0.05, 2, 1.5, 0.04877) MLE

BS

α

0.69794

0.14711

β

0.02173

0.00250

γ

0.31917

0.07057

R(t)

0.25340

0.08046

For β and γ, the symmetric Bayes estimators are more efficient than the asymmetric Bayes estimators. Not only that but, their estimated risks rather increase as c increases for fixed d.

5. Concluding Remarks In this paper, we develop Bayes estimators in the context of record statistics values from the exponentiated inverse Weibull distribution. Given non-informative prior distribution for β, it is not clear whether posterior distribution is proper or not. Therefore, we consider joint conjugate prior distribution used by Soland (1969). Using this prior, we can guarantee the existence of Bayes estimators as well as avoiding computational complexities of a joint prior. We derive the Bayes estimators for unknown parameters and reliability function R(t) using Soland’s method. Their corresponding MLEs are also obtained. The MLEs are compared with Bayes estimators based on the symmetric and two types asymmetric loss functions in terms of estimated risks. Our results show that the Bayes estimators superior to the MLEs under the informative (conjugate) prior. Specially, the asymmetric Bayes estimators are generally better than the symmetric Bayes estimators provided using a suitable value of c and d.

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Chandler, K. N. (1952). The distribution and frequency of record values, Journal of the Royal Statistical Society, Series B, 14, 220–228. Dumonceaux, R. and Antle, C. E. (1973). Discrimination between the lognormal and Weibull distribution, Technometrics, 15, 923–926. Mahmoud, M. A. W., Sultan, K. S. and Amer, S. M. (2003). Order statistics from inverse Weibull distribution and associated inference, Computational Statistics & Data Analysis, 42, 149–163. Maswadah, M. (2003). Conditional confidence interval estimation for the inverse Weibull distribution based on censored generalized order statistics, Journal of statistical Computation and Simulation, 73, 887–898. Nelson, W. B. (1982). Applied Life Data Analysis, John Willey & Sons, New York. Soland, R. M. (1969). Bayesian analysis of Weibull process with unknown scale and shape parameters, IEEE Transaction on Reliability, 18, 181–184. Soliman, A. A., Abd Ellah, A. H. and Sultan, K. S. (2006). Comparison of estimates using record statistics from Weibull model: Bayesian and non-Bayesian approaches, Computational Statistics & Data Analysis, 51, 2065–2077. Sultan, K. S. (2008). Bayesian estimates based on record values from the inverse Weibull lifetime model, Quality Technology & Quantitative Management, 5, 363–374. Thompson, R. D. and Basu, A. P. (1996). Asymmetric loss function for estimating system reliability, In Bayesian Analysis in Statistics and Econometrics, edited by Berry, D. A., Chaloner, K. M., and Geweke, J. K., Willey, 471–482. Varian, H. R. (1975). A Bayesian approach to real estate assessment, In Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage, edited by S. E. Feinberg and A. Zellner, North Holland, Amsterdam, 195–208. Zellner, A. (1986). Bayesian estimation and prediction using asymmetric loss function, Journal of American Statistical Association, 81, 446–451. Received March 7, 2012; Revised March 22, 2012; Accepted April 24, 2012